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Difference between revisions of "2015 IMO Problems/Problem 3"

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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
 
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
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==Solution==
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{{solution}}
  
 
==See Also==
 
==See Also==
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[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]
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Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
−
 
−
Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.Let <math>ABC</math> be an acute triangle with <math>AB>AC</math>. Let <math>\Gamma</math> be its circumcircle, <math>H</math> its orthocenter, and <math>F</math> the foot of the altitude from <math>A</math>. Let <math>M</math> be the midpoint of <math>BC</math>. Let <math>Q</math> be the point on <math>\Gamma</math> such that <math>\angle HKQ=90^\circ</math>. Assume that the points <math>A</math>, <math>B</math>, <math>C</math>, <math>K</math>, and <math>Q</math> are all different, and lie on <math>\Gamma</math> in this order.
 
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Prove that the circumcircles of triangles <math>KQH</math> and <math>FKM</math> are tangent to each other.
 

Latest revision as of 01:34, 19 November 2023

Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HKQ=90^\circ$. Assume that the points $A$, $B$, $C$, $K$, and $Q$ are all different, and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Solution

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See Also

2015 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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